Some groups whose reduced \(C^*\)-algebra is simple
Group \(C^*\)-algebras play vital roles, both in \(C^*\)-algebra theory and in harmonic analysis on locally compact groups. Therefore, the question of when the reduced \(C^*\)-algebra of a discrete group \(\Gamma\), \(C^*_r(\Gamma)\), is simple is of significance and has long been a matter of interest. The first milestone in the area was Powers' theorem that if \(\Gamma\) is a non-abelian free group, then \(C^*_r(\Gamma)\) is simple and has a unique normalized trace. Subsequently, this has been generalized by several authors. In the paper under review the authors obtain further important results. Let \(G\) be a connected real semisimple Lie group without compact factors and with trivial centre, and let \(\Gamma\) be a lattice in \(G\). The authors verify the long standing conjective that \(C^*_r (\Gamma)\) is simple (and has a unique normalized trace). Actually, they prove a considerably stronger result. Namely, let \(G\) be as above, suppose that \(\Gamma\) is a Zariski dense subgroup of \(G\) and endow \(\Gamma\) with the discrete topology. Then \(C^*_r (\Gamma)\) is simple and has a unique normalized trace (Theorem 1). Notice that by Borel's density theorem, Theorem 1 in particular applies to lattices. Moreover, Theorem 1 implies the following extension of a result of Howe and Rosenberg. Let \(G\) be a connected semisimple algebraic group defined over a field \(k\) of characteristic zero and with trivial centre. Let \(\Gamma\) be the group of \(k\)-rational points of \(G\), equipped with the discrete topology. Then \(C^*_r(\Gamma)\) is simple. In addition, there are a generalization of Theorem 1 to reduced crossed product \(C^*\)-algebras as well as some other interesting results. A crucial property for the discrete group \(\Gamma\) studied in the paper, which implies simplicity of \(C^*_r (\Gamma)\), is the following property \((P)\). A discrete group \(\Gamma\) with left regular representation \(\lambda_\Gamma\) is said to have property \((P)\) if given any finite subset \(F\) of \(\Gamma \setminus \{e\}\), there exist \(x \in \Gamma\) and a constant \(c> 0\) such that \(|\sum^\infty_{j = 1} a_j \lambda_\Gamma (x^{-j} yx^j)|\leq c |a|_2\) for all sequences \(a = (a_j)_{j \in \mathbb{N}}\) in \(l^2 (\mathbb{N})\) and all \(y \in F\).
reduced crossed product \(C^*\)-algebras
scientific article; zbMATH DE number 753835
scientific article